Imagine a straight tunnel directly through the center of the Earth. Now imagine an unpowered vehicle attached to rails in this tunnel. You get in this vehicle and to start your journey you release the clamps and begin to fall by the power of gravity alone. As you fall toward the Earth's core, you accelerate. Then as you pass the Earth's core, you start to decelerate. It works out (if you ignore wind resistance) that when you reach the surface of the Earth on the other side of the tunnel, you will come to a stop (and engage the clamps to prevent falling back in).

How long would that complete journey take? 42 minutes.

Now imagine a straight tunnel from Los Angeles to New York. This tunnel would cut through the mantle at a shallow angle. A journey through this tunnel would also take 42 minutes in a vehicle powered by gravity alone.

It turns out that a tunnel between any two points on the globe would take 42 minutes to traverse.

You're right that the acceleration is not uniform -- as you move closer to the center of the earth, acceleration decreases linearly. This is similar to harmonic oscillation, e.g. a mass hanging from a spring.

Turns out you can work the problem the same way by imagining a frictionless plane tangent to the earth's surface onto which you place a mass. Regardless of how far from the actual tangential point on the plane that you place the mass, it will take 42 minutes to make half an oscillation.

Sometimes I miss doing physics.

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"I wish I had documented more…" said nobody on their death bed, ever.

better yet, We could instal massive generators running off the earths core and have monsterously powerful magnetic motors that will shoot you through at the speed of light then when you get to the other side you stop. Really fast.

Hmm.. I guess 42 is pretty coo. By my calculations we could have something like that built in 42 years and 42 days and at the cost of only 42 trillion dollars. The total journey would only take you .042 seconds and you can only ride it 42 times before your face falls off from the extereme acceleration and deceleration.